CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Effective program specialization

Statement

Fix an acceptable numbering. For every index $e$ of an $(m + n)$ -ary program and every fixed parameter tuple $(a_{1}, \dots, a_{m})$ , one can effectively compute an index for the residual $n$ -ary program obtained by hard-wiring those parameters.

Facts & Assumptions

Given: An acceptable numbering, an index $e$ , and fixed parameters $a_{1}, \dots, a_{m}$ .

[L1]

For every $m, n$ , the $s$ -m- $n$ theorem gives a total computable specialization map $s_{m}^{n}$ , by The s-m-n theorem.

Proof

technique · direct

1.1

Apply [L1] to the chosen pair $(m, n)$ . The number $e^{'} : = s_{m}^{n} (e, a_{1}, \dots, a_{m})$ is effectively computable from the data.

L1givenconstruct

2.1

By the defining property of $s_{m}^{n}$ , the indexed function $φ_{e^{'}}$ agrees on every remaining input tuple with the original program $φ_{e}$ after the first $m$ inputs have been fixed to $a_{1}, \dots, a_{m}$ . Thus $e^{'}$ is the desired residual-program index.

step 1.1

3.1

Therefore program specialization is effective.

step 2.1∎

Depends on

The s-m-n theorem

Used by

Specializing a two-argument program to obtain a unary residual program Example

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lawrence S. Moss, Invitation to Computability and Recursion, The s-m-n Theorem (standard reference, not scraped)